94 
10. We may now prove as a consequence of the last 
remark that a pure sixfold statement either contains a 
group of four with a pair oddly distant from it or consists 
of two triads oddly distant from one another. For there 
must be a pair at distance 2 ; if the other four are all oddly 
distant from these, they form a group ; if one is evenly dis- 
tant and three oddly distant, we have the case of the two 
triads ; if two are evenly distant, we again have a group. 
We must add then first to a proper group and then to an 
improper group a pair oddly distant from it. To a proper 
group, consisting of the proximates to a certain origin, we 
may add the origin or its obverse with a mediate, or two 
mediates; S types. An improper group is symmetrical; 
that is to say, if we substitute for any one of its marks the 
obverse of that mark, we shall obtain a proper group. In 
this way we shall get four origins, distant 1113 from the 
group, and four obverses distant 1333 ; if we add to these 
the obverses of the marks in the group itself, we have de- 
scribed the relation of the twelve remaining marks to the 
group. To form therefore a pure six-fold statement we may 
add either two origins or two obverses or an origin and an 
obverse ; 3 types. 
In the case of the two triads, since they are oddly distant 
from one another their origins must be oddly distant, that 
is, they must be distant either 1 or 3. If they are distant 
1 , neither, both, or one of the origins may appear in the 
statement ; if they are distant 3, neither, both, or one of the 
obverses; 6 types. Thus we obtain 12 types of purely six- 
fold statement. 
11. If a six-fold statement contains one pair of obverses, 
the remaining four marks cannot all be evenly distant from 
this pair. For in that case they would constitute a group ; 
and it is easy to see that the marks evenly distant from a 
group, whether proper or improper, do not contain a pair of 
obverses. W e have therefore only these four cases to consider : 
